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I'm conducting a clinical study where I determine an anthropometrical measure of the patients. I know how to handle the situation where I have one measure per patient: I make a model, where I have a random sample $X_1,\dots,X_n$ from some density $f_\theta$, and I do the usual stuff: write the likelihood of the sample, estimate parameters, determine confidence sets, and test hypothesis, or even do some Bayesian analysis if the boss isn't watching. ;-)

My problem is that for some patients we have more than one measure, because we believe that it is a good idea to have more than one researcher handling the measuring device, when this is possible (some times we have just one researcher working at the clinic). Therefore, for some patients we have one measure made by one researcher, for other sample units we have two measures made by two different researchers, and so on. The measure in question is the thickness of a specific skin fold.

My question: which kind of statistical model is adequate for my problem?

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    $\begingroup$ Unless the questions of interest relate to the specific researchers, and you have the information to tell who made which measurements, you might be looking at mixed models, with a 'researcher' random effect (which might account for things like a researcher having small but consistent biases, for example). $\endgroup$
    – Glen_b
    Commented Sep 1, 2013 at 1:09
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    $\begingroup$ Have you the information of which researcher did which measurement? Do you think that some researches do systematic errors? Or that some researchers measure more precisely than others? $\endgroup$
    – user31264
    Commented Oct 31, 2013 at 7:51
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    $\begingroup$ In practice, do you actually find large differences among researchers measuring the same patient at about the same time? $\endgroup$
    – EdM
    Commented Sep 15, 2014 at 17:29
  • $\begingroup$ Just as @Glen_b I think you should look at mixed effects models, maybe this can be a start: stats.stackexchange.com/questions/166434/… $\endgroup$
    – user83346
    Commented Aug 26, 2015 at 14:53

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Take a look at paper of Brennan (1992) on Generalizability Theory or his book, also titled "Generalizability Theory" (2010, Springer). Brennan writes about GT using ANOVA, but mixed models could be used the same way - and many would consider them as a more recent method.

You could think of a mixed model for cross-classified data (e.g. Raudenbush, 1993). Say you have $N$ patients measured by $R$ researchers, and your measurement is denoted as $X_{ij}$ for $i = 1,...,N$ and $j = 1,...,R$. In this case, the measurement has both effects of patients and researchers, with patients "nested" in researchers (multiple measures for a single patient) and researchers "nested" in patients (multiple measurements for each patient), so

$$ X_{ij} = \beta_0 + b_i + b_j + \varepsilon_{ij} $$

where $\beta_0$ is an fixed intercept (if the data is not centered), $b_i$ is patient random effect (random intercept) and $b_j$ is a researcher random effect, while $\varepsilon_{ij}$ is an error term. In lme4 this would be

x ~ (1|patient) + (1|researcher)

you could extend this approach to using $X$ as an independent variable or define a hierarchical Bayesian model where you include both sources of variability.

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I will take a poke at this even though I can only provide a mathematical model, as I am a bit of a math nerd, but not a statistician.

Kalman Filters can handle state estimation with multiple-inputs and missing information.

If I had to show this to engineers, they would require me to make variability plots of measures between measuring-technicians to show there is no operator-to-operator variability. They would treat two measurements as paired. Stats folks are good at this. If the operator-to-operator variability was negligible then I could formulate my data with each as a single line.

  • [... measurement_1 ... result]
  • [... measurement_2 ... result]

if only one technician made the measurement there would be only one line of data

otherwise, I would want to have an indication of operator within the data

  • [... operatorname measurement ... result]

If you can characterize the difference each operator has on the same measurement, then you can account for it in your model. If you don't supply an indicator of operator, when it is a significant source of variability ... that could be a problem.

The data model informs the mathematic model. I think GLM's have had good results in these areas. http://www.uta.edu/faculty/sawasthi/Statistics/stglm.html

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I am also coming at this question from a different field. Regardless, it sounds to me like the purpose of having multiple people use the measuring device is to be able to account for measurement error? If I am correct in my understanding of what you are trying to do, then it sounds like a case for structural equation modeling (SEM), which would allow you to run your model free of measurement error. SEM can account for missing data if you use FIML estimation techniques, you have to make the usual assumptions about the missing data (i.e., at least missing at random). SEM models have been used increasingly in RCT settings, so I don't think it would be uncommon to use this technique. The question I'd have is: do you have enough information to make a properly identifiable SEM model?

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